# Physics 9702 Doubts | Help Page 63

__Question 353: [Radioactive decay]__Because of the random nature of the radioactive emission process, the count-rate recorded by a Geiger-Muller tube or solid-state detector from a supposedly constant source is subject to statistical fluctuations. For example, if the total count recorded from such a source in a given time is N, the variability of repeated measurements is of the order of √N. For how many counts should counting continue in order to obtain the mean count rate to a precision of about 1%?

A 100 B 1 000 C 10 000 D 100 000 E 1 000 000

**Reference:**

*Past Exam Paper – N83 / II / 35*

__Solution 353:__**Answer: C.**

The variability of repeated measurements (ΔN) is of the order of √N.

(√N / N) x 100% = 1%

√N / N = 0.01

N

^{0.5}N

^{-1}= N

^{-0.5}= 0.01

1 / √N = 0.01

N = (1 / 0.01)

^{2}= 100

^{2}= 10 000

__Question 354: [Nuclear Physics]__A stationary

^{238}U nucleus decays by α emission generating a total kinetic energy T.

^{238}

_{92}U - - - >

^{234}

_{90}Th +

^{4}

_{2}α

A slightly less than T / 2

B T / 2

C slightly less than T

D T

E slightly greater than T

**Reference:**

*Past Exam Paper – N 90 / I / 6*

__Solution 354:__**Answer: C.**

The

^{238}U nucleus is initially stationary, so it has zero momentum.

From the law of conservation of momentum, the sum of momentum before the decay should be equal to the sum of momentum after the decay.

After the decay,

Let momentum of

^{234}

_{90}Th = Mv

_{Th}= (234u) v

_{Th}

and let the momentum of

^{4}

_{2}α = mv

_{α}= (4u) v

_{α}

From the law of conservation of momentum,

(234u) v

_{Th}+ (4u) v

_{α}= 0

Considering the magnitudes,

(234u) v

_{Th}+ (4u) v

_{α}

Speed of α-particle, v

_{α}= (234u) v

_{Th}/ (4u)

So, the speed of alpha-particle is much greater than that of Th. Since kinetic energy depends on the square of speed, the kinetic energy of the α particle is then just slightly less than T.

__Question 355: [Simple harmonic motion]__A particle undergoes simple harmonic motion on a straight line with amplitude 5.0 cm and frequency 2.0 Hz. At time t = 0, it starts to move to the right from the equilibrium position. Determine

**(a)**the displacement at t = 0.10 s and t = 0.40 s;

**(b)**the instants when the displacements of the object are 1.0 cm and – 2.0 cm for the first time.

__Solution 355:__The equation describing the displacement x of a particle undergoing simple harmonic motion with respect to time t is given by

x = x

where x_{0}sin (ωt)_{0}is the amplitude (maximum displacement) (= 5.0cm) and ω is the angular frequency of the simple harmonic motion

Angular frequency ω = 2π /T = 2πf = 2π (2) = 4π

So, the equation can be simplified as

x = 5 sin (4πt)

Note that the argument of the sine function should be in radians.

**(a)**

x = 5 sin (4πt)

When time t = 0.10s,

Displacement x = 5 sin (4π (0.10)) = 4.76cm

When time t = 0.40s,

Displacement x = 5 sin (4π (0.40)) = – 4.76cm

**(b)**

x = 5 sin (4πt)

To find the time, we need to make t the subject of formula.

sin (4πt) = x / 5

4πt = sin

^{-1}(x/5)

Time t = [sin

^{-1}(x/5)] / 4π

When displacement x = 1.0cm

Time t = [sin

^{-1}(1.0/5)] / 4π = 0.016s = 0.02s

When displacement x = – 2.0cm

Time t = [sin

^{-1}(– 2.0/5)] / 4π = 0.016s = 0.28s

__Question 356: [Electricity > Electric field strength]__
Two large flat metal plates A and B
are placed 9.0 cm apart in a vacuum, as illustrated.

A potential difference of 450 V is
maintained between plates by means of a battery.

**(a)**

(i) On Fig, draw arrow to indicate
the direction of the electric field between plates A and B.

(ii) Calculate electric field
strength between A and B.

**(b)**An electron is released from rest at surface of plate A.

(i) Show that change in electric
potential energy in moving from plate A to plate B is 7.2 ×10

^{–17}J.
(ii) Determine speed of the electron
on reaching plate B.

**(c)**On axes of Fig, sketch graph to show the variation with distance d from plate A of the speed v of the electron.

**Reference:**

*Past Exam Paper – November 2003 Paper 2 Q5*

__Solution 356:__**(a)**

(i) The arrow is from B towards A.

(ii) Electric field strength, E = V
/ d = 450 / (90x10

^{-2}) = 5.0x10^{3}NC^{-1}**(b)**

(i)

Change in electric potential energy
= qV or Eqd

Change in electric potential energy
= (1.6x10

^{-19}) (450) = 7.2x10^{-17}J
(ii)

{The change in electric
potential energy is converted into kinetic energy of the electron.}

Kinetic energy, E

_{k}= ½ mv^{2}
7.2x10

^{-17}= 0.5 (9.1x10^{-31}) v^{2}
Speed v = 1.26x10

^{7}ms^{-1}**(c)**

The line starts from the origin, and
is curved in the correct direction, but not ‘level out’

Salam!

ReplyDeletein the formula,d sin θ = n λ, for a diffraction grating, what is "n"?

in the same formula for Young's double slit setup, is "n" the order?

in the formula, "d=1/n", is "d" the distance of separation of two slits and is "n" the number of slits/lines per unit length of the grating?

Wslm.

DeleteFor the diffraction grating equation: d sinθ = nλ, n is the n^th order of diffraction visible and d is the slit separation.

For example, consider question 381 at

http://physics-ref.blogspot.com/2015/02/physics-9702-doubts-help-page-69.html

For a double slit case, consider question 379 on the same reference given above.

AOA. For Q 356 part c, I don't get the concept behind the graph, even though its just for one mark, but what i've understood is that as the distance between the plates increases, the speed decreases, but why can't it ever level out/become constant? And isn't speed just connected to the Voltage not the separation between the plates? Or is it like if the Electric Field strength is constant, the distance and voltage are directly proportional? but in this case if the distance is increasing, the field strength has to change right.. oh God everthings so messed up in my head :S

ReplyDeleteThe electric force causes a constant acceleration on the electron.

DeleteAn acceleration means that the velocity increases, it does not remain constant.

Thats correct, and due to constant acceleration provided by the electric force the velocity uniformly keeps increasing and so for every other interval of velocity the distance traveled increases ie: the distance traveled while the velocity increases from 5 to 10 is more than the distance traveled while velocity is increasing from 0 to 5.

Deleteif acceleration is constant then shouldn't the graph be linear ?

DeleteNo, because this is a velocity-distance graph, not a velocity-time graph. See explanation in comments above.

Delete